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Note: Yoy W; 8 rank Frobenius 3 norm 2 IF(a) Find the (6 points) 8 & & € closest E & € & singular (with ja233 respect E & € &a...

Question

Note: Yoy W; 8 rank Frobenius 3 norm 2 IF(a) Find the (6 points) 8 & & € closest E & € & singular (with ja233 respect E & € & the decomposition of a Frobenius 20 115 1 (Wou 9068 1 0.8 of rank follows:

Note: Yoy W; 8 rank Frobenius 3 norm 2 IF (a) Find the (6 points) 8 & & € closest E & € & singular (with ja233 respect E & € & the decomposition of a Frobenius 20 115 1 (Wou 9068 1 0.8 of rank follows:



Answers

If a $3 \times 8$ matrix $A$ has rank $3,$ find $\operatorname{dim} \mathrm{Nul} A, \operatorname{dim} \operatorname{Row} A$ and $\operatorname{rank} A^{T} .$

And this problem we are given that Matrix A. is the three x 2. Matrix six I get 1- zero. Now you have 34. Mhm. And Matrix B is the Matrix Uh huh. 31 And yet 1560. And we want to find first A plus B. Now to add to major cities, you just add the corresponding components together. So top left close, Top left Is what goes on. The top left search plus three is 9. Top ride was top ride, It was in the top right now give one postponement 0. And then we proceed in this matter. Two plus negative one is one, zero Plus 5 is five. They have three plus six is three Than 4.0. As for And so that is our Matrix A plus B. Now for the Matrix A minus B. Take our correspondent components again and subtract the light terms 6 -3 is three -1 -1 is -2. Two minus negative one is three, 0 -5 is -5. Then you have 3 -6 is -9. 4 0 is four. Now for two a. We take everything and a and multiply it by two. six times 2 is 12 in one times two is that you have to We have 40 -6 and eight. And then lastly we want the matrix negative three C. I'm sorry, 93 B. Three B. That means take everything and be and multiply it by negative three 789 Now you have three. Three may have 15 negative 18 zero.

The problem we're told the matrix A. Is equal The 3 -22 zero on negative four. And negative three to negative form. And the Matrix B is a matrix 40 two negative one negative 13 Now first we want to find A plus B. Now in order to add two matrices they must have the exact same dimensions. Well a is a three x 3 makers But B is a three x 2. And so that means that this is not possible because they don't have the same dimensions. Yeah. We also want to find A minus B. Yeah this is something else that we can't do because these don't have the same dimensions. And so this is still not possible. Yeah. Now we can find to A to A. Just means take everything in A. And multiply it by two. So that top probably six. And then get a 44 Than 0 to -8. The -6 4 -2. Yeah. And then lastly we want negative three B. And so it just means to take everything and be And multiply it by -3. That's 1912 0. Then you have 63 3 -9. So that's our matrix -3 B.

I only have a matrix A. As the Matrix 0 -27 five four, negative three. And be the matrons 840 zero 14 Now first thing we want to find here is a plus B. Finding a Polsby just means add the parts that go together. So I had the top left with the top left, the middle with the middle, the top right with the top right. And then so on zero plus eight gives us eight. They have two was forced to 707 On the bottom 505 four Plus 1 is five. Three plus four is one. So that's our major. It's A plus B. No a minus B is very similar. So if we subtract instead So in the top left, zero mindset is -8. They have to -4 is negative soups 7 -07. On the bottom 5 0 is five. 4 -1 is three. And I give 3 -4. There's negative stuff not for the matrix to A. That just means take every component in a multiplied by two. Mhm. And so this becomes zero. Now you're four 14 10, eight -6. That's our matrix to a. We have three. B is very similar to just take everything and be multiply it by -3. So it's 1924. May have 12 zero zero nee of three. No, you have 12.

Okay, So what we have here is supported by board meat. Thanks. When we're asked the fine, it's reduced row, excellent form. And it's round. So based from our previous nous, since we need to convert disperse with its throw at you. And for that I mean it's strong and no further operations in Britain introduced row for So this your instrument for muses deceive elementary row operations We throw incident for but this different great beer. Yeah, So we say that, uh, Matrix has seen its reduced ruined student for if it's stopping spice the four requirements first, the 1st 90 number in the first row is number. What set one? The second row also starts with the number one, which is further to the right than the leading enter in the first row. So for every subsequent roll, the number one must be further to the right, so it should look like this. Resembling as thirties. It's so three, the leading Andrian each room must be the only non zero in its Kahlo. And finally, and in an zero, those are placed at the bottom six. So we can do this by going serious. A parliamentary row operations, including in their changing wonder with another multiplying one drove and non zero Pakistan and replacing one room with one row plus a constant times another rope. So for this example, what we're going to do first is to transform it into its role. Excellent For so our first step is to swap Are one is this what are one? We'd our port. So by doing that, we know have for negative too. 30 eight, three negatives, too. 07 Cool. Negative one Oh, four and one negative. One negative one. Next they want to cancel the leading efficient in art. And we do this by doing arto. Sorry, we do this by doing are too minus three ports so far. What? By doing that, we know how for negative too. But he eight zero They get the one house negative 9/4 one. So negative one Oh, four and one negative One negative one. Next we want to cancel. The leading coefficients are three and we do this by doing our three minus one help off our one With that, we now have four Negative too. 30 eight zero negative one house negative. 9/4 one zero zero one have zero and finally one negative one Negative. What? For our fourth step, I have your stand following. What we want to do now is to cancel the leading coefficient in our port. So we do this by building are for minus 14 Fire went now we have for blanketing too. Three e zero negative. 1/2 negative. Mine over four and one 00 1/2 zero and then zero. Then get the 1/2 negative. 7/4, then zero. Next again, we want to cancel the leading coefficient off our port, which is negative Would happen. So what we're going to do now is are for bind us are too. And with that, we now have four negative too. 30 eight zero. Get the 1/2 negative. Nine Gilbert four one 00 1/2 zero and then zero zero one. Have they get the one our six is to and then cancel the leading coefficient in row four. So by cancelling that peace will satisfy one off art criteria in row insurance form. So we do this now by are four by doing art for minus are the Now we have for negative too Pretty eight zero. They got the 1/2. They get the 9/4. One zero zero went help. Zero, then 000 Negative. What? So this is already on its row? Answered on form. And what we want to do now is to get drunk. So by definition, the run is the maximum number off linearly independent, better in a matrix. So this is equal to the number off vans era rose in its euro insolent meetings. So apparent a look at the example. We have 12 and three rows with non zero element, which is roll one road boom and there are three. Therefore, our room it's three. So we're not quite done yet since we need to get the reduce row petulant form off our matrix. So what they want to do now first is to is to divine are one we'd for. So now we get one Megat the 1/2 but imports zero negative 1/2 negative nine bill before one 00 one house zero and then 000 negative. What next? We want to me zeros in column toe except at the entry of pro to with that What we're going to do is to Centra are who you are. What we now have one zero 30 one zero negative 1/2 Leggett The night Cooper for one zero zero. When have zero then 00 zero negative. What? So we're almost done with our solution. So I hope yourself follow we Our next step is to multiply. Art do well, Negative that we know have one zero 30 one zero one nine over to and negative do 00 1/2 zero and then 00 zero negative. What? So for our next there, What we want now is to make zeros in column three and step at the entry of three. So we don't this by subtracting six are three who are one with death. We now have 1001 zero one 9/2 negatives do. 00 1/2 zero 000 Negative. What? Oops. X over and now this is Jay. This is I'm just gonna rewrite businesses. Step que who? Not those Any confusion. So we're now at step ill and what we're going to go now is to said nine rt Who are you? So we now have 1001 zero, 10 Negative too. 00 1/2 0000 What? Or step em. What we want to do now is the multiply our city. We thought so We get one 001 010 Negative 00 10 then 00 zero Negative four step and we're almost done. What we want to do now is to make zeros in column for accepting the entire approach. For So we do this by Oops, I'm being are one. And I report No, we have 1000 010 Negative, too. 0010 and 000 Negative. So far. Step. Oh, we're going to set it up now. Two off are for so I do. Now we have one 000 010 The room 0010 000 Negative. What? And for our final step, what we're going to do now is to multiply our for with negative one. And what we're going to get is 1000 0100 0010 and 000 What? So this is now that reduce role? Excellent form off the medics


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